Method and apparatus for high-order PAPR reduction of an OFDM signal

ABSTRACT

A method and apparatus for high-order peak-to-average power ratio reduction of an OFDM signal are disclosed. The method partitions time-domain input data x[n] of length N into M disjoint subblocks in time domain, and a complete N-point transmitted signal {tilde over (x)}[n], n=0, 1, . . . , N−1, is composed after transformation, complex multiplication, and phase optimization, where M is a power of 2, M≧8 and N/M&gt;1 is an integer. Accordingly, the apparatus comprises an N-point inverse fast Fourier transform (N-IFFT), a de-multiplexer, a transformer, two sets of memories, a plurality of complex multipliers, and an adder. This invention uses only one N-IFFT, whereby it achieves significant computation reduction. As M=8, the number of complex multiplications and that of memory units required are less than or equal to (N/2)log 2 N+(3N/4) and 3N/2, respectively. The invention also preserves the inherent property as well as advantages of an OFDM system.

FIELD OF THE INVENTION

The present invention generally relates to a method for solving theproblem of peak-to-average power ratio (PAPR) of an orthogonal frequencydivision multiplexing (OFDM) signal at transmission side, and morespecifically to a method for high-order PAPR reduction of an OFDMsignal, and an apparatus of performing the same.

BACKGROUND OF THE INVENTION

Multicarrier communication systems such as discrete multitone (DMT) andOFDM systems have attracted much attention in the applications ofhigh-speed communication. The advantages of the multicarriercommunication system lie in partitioning a high speed data stream into aplurality of parallel data streams, each transmitted by a subcarrier. Assuch, each data stream is transmitted at low speed, and thus has astronger capability in anti-multipath channel effect and narrow bandinterference.

FIG. 1 shows a block diagram of a conventional OFDM transmitter. In theOFDM transmitter, the input data X[k], k=0, 1, . . . , N−1, aretransmitted in an OFDM symbol period, where N is the number of usefuldata in an OFDM symbol. After the serial/parallel transformation,N-point inverse fast Fourier transform (N-IFFT), and parallel/serialtransformation, the input data are transformed into the followingdiscrete time sequence:

$\begin{matrix}{{{{x\lbrack n\rbrack} = {\frac{1}{\sqrt{N}}{\sum\limits_{k = 0}^{N - 1}{{X\lbrack k\rbrack}W_{N}^{kn}}}}},\mspace{14mu}{n = 0},1,\ldots\mspace{11mu},{N - 1}}{where}} & (1) \\{W_{N} \equiv {\mathbb{e}}^{{j2\pi}/N}} & (2)\end{matrix}$is the twiddle factor. The discrete time sequence x[n] obtained fromequation (1) undergoes the cyclic prefix insertion and digital/analogtransformation to obtain an analog signal x(t). The analog signal x(t)is then transmitted to the RF front end for further processing,including an IQ modulation, an up conversion, and a power amplification.The PAPR of the analog signal x(t) is several dB higher than the PAPR ofthe corresponding discrete time sequence x[n], and is close to the PAPRof x[n/R]. where x[n/R] represents the sequence obtained by R timesoversampling of x[n]. Therefore, the PAPR of x(t) can be approximated byusing x[n/R] as follows:

$\begin{matrix}{{PAPR} = \frac{\max\limits_{0 \leq n \leq {{RN} - 1}}{{x\left\lbrack {n/R} \right\rbrack}}^{2}}{E\left\{ {{x\left\lbrack {n/R} \right\rbrack}}^{2} \right\}}} & (3)\end{matrix}$Where E{·} is the expectation operation. The approximation is relativelyaccurate when R≧4. However, one of the main disadvantages ofmulticarrier communication systems is the high PAPR of the modulatedsignal. When the modulated signal with a high PAPR passes through the RFfront end, the signal is distorted due to the non-linearity of a regularRF amplifier. The non-linearity not only causes the in-band signaldistortion which leads to the increase of the bit error rate (BER), butalso causes the out-of-band radiation which leads to the interference ofadjacent channels, a violation of the government regulation. Astraightforward solution to this problem would be using an RF amplifierwith a larger linear range. However, the aforementioned solution willlead to the reduction of power efficiency, higher power consumption anda higher manufacturing cost.

There are several conventional methods for solving the aforementionedproblem. Among these methods, the partial transmit sequences (PTS) ismost attractive due to its relatively low realization complexity andcapability in PAPR reduction. Ericsson (U.S. Pat. No. 6,125,103)disclosed a method for using PTS to solve the high PAPR of the signal atthe OFDM transmission end, as shown in FIG. 2. The explanation is asfollows.

First, the input data X[k] of length N is partitioned in the frequencydomain into M disjoint subblocks, represented by X₁[k], X₂[k], . . . ,X_(M)[k], k=0, 1, . . . , N−1. The partition can be interleaved,adjacent, or irregular, as shown in FIG. 3 (using M=8 as an example).The M disjoint subblocks are phase-rotated and added to form thefollowing signal:

$\begin{matrix}{{{\overset{\sim}{X}\lbrack k\rbrack} = {\sum\limits_{m = 1}^{M}{b_{m}{X_{m}\lbrack k\rbrack}}}},\mspace{14mu}{k = 0},1,\;\ldots\mspace{11mu},{N - 1}} & (4)\end{matrix}$where b_(m) is the phase rotation parameter of the m-th subblock (mε{1,2, . . . , M}) and |b_(m)|=1.Equation (4), after the N-IFFT, becomes:

$\begin{matrix}{{{\overset{\sim}{x}\lbrack n\rbrack} = {\sum\limits_{m = 1}^{M}{b_{m}{x_{m}\lbrack n\rbrack}}}},\mspace{14mu}{n = 0},1,\ldots\mspace{11mu},{N - 1}} & (5)\end{matrix}$where x_(m)[n] is the result of the N-IFFT of the X_(m)[k]. In the PAPRreduction, the object of the PTS method is the phase optimization, i.e.,seeking for the optimal sequence {b₁, b₂, . . . , b_(M)} so that thePAPR of the transmitted signal is minimum. In practice, the phase ofb_(m) is usually restricted to one of the four possibilities {+1, −1,+j, −j} so that no multiplication operation is required in the phaserotation.

From FIG. 2, it can be seen that an N-point OFDM symbol requires M timesof N-IFFT operation. That is, a total of M·(N/2)log₂ N complexmultiplications are required. Several methods are further devised toreduce the amount of computation required in the PTS method. Kang, Kimand Joo, in their article “A Novel Subblock Partition Scheme for PartialTransmit Sequence OFDM,” IEEE Trans. Broadcasting, vol. 45, no. 3, pp.333-338, September 1999, disclosed a method of using the characteristicsof the PTS interleaved partition of the subblocks, as shown in FIG. 4(M=8). Each subblock has N points in the frequency domain, but only Lpoints of them have non-zero values (L=N/M). Therefore, the N-IFTT onthe N-point subblock X_(m)[k] is equivalent to the L-IFFT on the L-pointsubblock (where X_(m)[k] has non-zero values), repeating M times in thetime domain to form the N-point signal, and multiplying the N-pointsignal with the N-point complex coefiicients:(1/M)·e^(j2πmn/N), m=0, 1, . . . , M−1, n=0, 1, . . . , N−1This method takes M·(L/2)log₂ L+MN multiplications, and requires MNunits of memory space.

Samsung (US. Patent publication 2003/0,067,866) disclosed a similarmethod, as shown in FIG. 5. The Samsung method differs from the previousmethod in no repetition after the L-IFFT on an L-point subblock.Instead, the multiplication of the L-point complex coefficients in thetime domain is performed to make the time domain subblocks orthogonal sothat the receiving end can separate each subblock. As there are only Lpoints in each time domain subblock, the PAPR is lower and, therefore,the PAPR of the transmitted signal after the phase rotation and theaddition is also lower. Although this method takes M·(L/2)log₂ L+Nmultiplications and requires N units of memory space, this methodreduces the length of the OFDM signal from N to L, which means that thecapability of anti-multipath channel effect is also greatly reduced.Furthermore, the L-point complex coefficient multiplier to make the timedomain subblocks orthogonal is hard to design and may not exist for mostapplications. This will further make the receiving end more difficult inobtaining the original transmitted data.

SUMMARY OF THE INVENTION

The present invention has been made to overcome the aforementioneddrawback of the conventional PTS methods for PAPR reduction of the OFDMtransmission end. The primary object of the present invention is toprovide a high-order PAPR reduction method and apparatus for the OFDMsignal.

The high-order PAPR reduction method according to the present inventionincludes the following steps: (1) using the N-IFFT to transform thefrequency domain signal X[k] of length N into the time domain signalx[n] of length N, where N is the number of useful data in an OFDMsymbol, and k, n=0, 1, . . . , N−1; (2) partitioning the time domainsignal x[n] of length N into M disjoint subblocks, each subblock havingthe length of N/M, M being a power of 2, M being greater or equal to 8,and N/M being an integer greater than 1; (3) transforming the Msubblocks into M sub-sequences z_(l)[n], each having the length of N/M,where l=1, 2, . . . , M, and n=0, 1, . . . , (N/M)−1; (4) usingfixed-phase rotation to rotate M sub-sequences z_(l)[n] and obtain Msub-sequences {tilde over (z)}_(l)[n], each having the length of N/M,where l=1, 2, . . . , M, and n=0, 1, . . . , (N/M)−1; and (5) using thephase rotation, phase optimization, and addition to obtain a completeN-point signal {tilde over (x)}[n] having the length of N.

Because not all the fixed-phase rotations require multiplier, thepresent invention of a high-order PAPR reduction apparatus for OFDMsignal includes an N-point inverse fast Fourier transform (N-IFFT), ade-multiplexer, two set of memories, a transformer, at most M complexmultipliers, and an adder. The N-IFFT transforms the input frequencydomain signal X[k] into the time domain signal x[n]. The de-multiplexeruses the adjacent partitioning to partition x[n] into M disjointsubblocks of identical length. The transformer transforms the M disjointsubblocks into M sub-sequences z_(l)[n] of length N/M, where l=1, 2, . .. , M and n=0, 1, . . . (N/M)−1. Some of the sub-sequences z_(l)[n] arecomplex multiplied and phase rotated, and the remaining sub-sequencesz_(l)[n] are directly phase rotated. Finally, the adder adds them toobtain a complete N-point signal {tilde over (x)}[n].

For explanation, the present invention uses a preferred embodiment and asimplified embodiment of M=8.

The present invention only uses one N-IFFT so that the amount ofcomputation is greatly reduced. In the preferred embodiment, the presentinvention takes (N/2)log₂N+(3N/4) complex multiplications and requires3N/2 units of memory space. In the simplified embodiment, the presentinvention takes (N/2)log₂N complex multiplications and requires N unitsof memory space. But more important, the present invention keeps theoriginal capability of anti-multipath channel effect in the OFDM system.

When M=8, the present invention shows different results in terms ofcomputation amount and memory requirement, compared to the other threePTS methods. Compared to the original PTS method and the methoddisclosed by Kang, Kim and Joo, the simplified embodiment of the presentinvention and the Samsung's method use the least memory space. While theamount of multiplication is slightly higher or equal to that required bythe Samsung's method, the present invention does not require shorteningthe length of the OFDM symbol, and therefore keeps the features andadvantages of the OFDM system.

The foregoing and other objects, features, aspects and advantages of thepresent invention will become better understood from a careful readingof a detailed description provided herein below with appropriatereference to the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a schematic view of a conventional multicarriercommunication system related to OFDM transmitter.

FIG. 2 shows a PTS method to solve the high PAPR problem of OFDM signalat transmission end.

FIG. 3 shows the three ways of partitioning the input data X[k] intosubblocks using M=8.

FIG. 4 shows the embodiment of the method proposed by Kang, Kim and Jooto reduce the computation amount of PTS.

FIG. 5 shows the embodiment of the method proposed by Samsung to reducethe computation amount and memory requirement of PTS.

FIG. 6 shows a schematic view of a multi-stage time domain signalpartitioning.

FIG. 7 shows a schematic view of 3-stage time domain signal partitioningwhen M=8.

FIG. 8 shows the high-order PAPR reduction method of OFDM signal of thepresent invention.

FIG. 9 shows the schematic view of the apparatus of the presentinvention for high-order PAPR reduction of the OFDM signal.

FIG. 10 shows an embodiment of the present invention in FIG. 9 when M=8.

FIG. 11 shows the transformer of FIG. 10.

FIG. 12( a) and FIG. 12( b) show the phase rotation parameter setting inFIG. 10.

FIG. 13 shows a simplified embodiment of the present invention when M=8.

FIG. 14 shows the transformer of FIG. 13.

FIG. 15( a) and FIG. 15( b) show the phase rotation parameter setting inFIG. 13.

FIG. 16 shows the comparison of the computation amount and the memoryrequirement between the present invention and the other three methods.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

FIG. 6 shows a schematic view of a multi-stage time domain signalpartitioning having m stages. The number of time domain signals afterpartitioning is M=2^(m). The N-IFFT is performed on the frequency domainsignal X[k] of length N to obtain a time domain signal x[n]. Assume theinitial value to be:y ₁ ⁽⁰⁾ [n]=x[n]  (6)At the first stage, the following equations can be used to partitionx[n] into two disjoint signals:

$\begin{matrix}\left\{ \begin{matrix}{{y_{1}^{(1)}\lbrack n\rbrack} = {{y_{1}^{(0)}\lbrack n\rbrack} + {y_{1}^{(0)}\left\lbrack \left( \left( {n - \frac{N}{2}} \right) \right)_{N} \right\rbrack}}} \\{{y_{2}^{(1)}\lbrack n\rbrack} = {{y_{1}^{(0)}\lbrack n\rbrack} - {y_{1}^{(0)}\left\lbrack \left( \left( {n - \frac{N}{2}} \right) \right)_{N} \right\rbrack}}}\end{matrix} \right. & (7)\end{matrix}$where the notation ‘((n))_(N)’ represents ‘(n modulo N)’, n=0, 1, . . ., N−1. In the same way, the two disjoint signals can be furtherpartitioned into four disjoint signals. Repeating this step until them-th stage, the following equations can be obtained for the M disjointsignals:

$\begin{matrix}\left\{ {\begin{matrix}{{y_{1}^{(m)}\lbrack n\rbrack} = {{y_{1}^{({m - 1})}\lbrack n\rbrack} + {y_{1}^{({m - 1})}\left\lbrack \left( \left( {n - \frac{N}{M}} \right) \right)_{N} \right\rbrack}}} \\{{y_{{({M/2})} + 1}^{(m)}\lbrack n\rbrack} = {{y_{1}^{({m - 1})}\lbrack n\rbrack} - {y_{1}^{({m - 1})}\left\lbrack \left( \left( {n - \frac{N}{M}} \right) \right)_{N} \right\rbrack}}}\end{matrix}\left\{ {\begin{matrix}{{y_{2}^{(m)}\lbrack n\rbrack} = {{y_{2}^{({m - 1})}\lbrack n\rbrack} + {{\mathbb{e}}^{j\frac{2\pi}{M}} \cdot {y_{2}^{({m - 1})}\left\lbrack \left( \left( {n - \frac{N}{M}} \right) \right)_{N} \right\rbrack}}}} \\{{y_{{({M/2})} + 2}^{(m)}\lbrack n\rbrack} = {{y_{2}^{({m - 1})}\lbrack n\rbrack} - {{\mathbb{e}}^{j\frac{2\pi}{M}} \cdot {y_{2}^{({m - 1})}\left\lbrack \left( \left( {n - \frac{N}{M}} \right) \right)_{N} \right\rbrack}}}}\end{matrix}\mspace{59mu}\vdots\left\{ \begin{matrix}{{y_{M/2}^{(m)}\lbrack n\rbrack} = {{y_{M/2}^{({m - 1})}\lbrack n\rbrack} + {{\mathbb{e}}^{j\frac{2{\pi{({\frac{M}{2} - 1})}}}{M}} \cdot {y_{M/2}^{({m - 1})}\left\lbrack \left( \left( {n - \frac{N}{M}} \right) \right)_{N} \right\rbrack}}}} \\{{y_{M}^{(m)}\lbrack n\rbrack} = {{y_{M/2}^{({m - 1})}\lbrack n\rbrack} - {{\mathbb{e}}^{j\frac{2{\pi{({\frac{M}{2} - 1})}}}{M}} \cdot {y_{M/2}^{({m - 1})}\left\lbrack \left( \left( {n - \frac{N}{M}} \right) \right)_{N} \right\rbrack}}}}\end{matrix} \right.} \right.} \right. & (8)\end{matrix}$This is the result of time domain signal partitioning required prior tothe PTS phase rotation.

Based on the multi-stage signal partitioning of FIG. 6, FIG. 7 shows aschematic view of M=8. According to the previous procedure, the 3-stagepartitioning for M=8 requires (N/2)log₂(N)+2N multiplications, and 8Nunits of memory. Based on the same concept, the following description ofa high-order PAPR reduction method for OFDM signals uses the symmetriccharacteristics of the M disjoint signals to lower the amount ofmultiplication and memory requirement.

Substituting equation (6) and the first stage equation (i.e., (7)) up to(m−1)th stage equation into equation (8), the M time domain inputsubblocks {x[n], n=0˜(N/M)−1}, {x[n], n=2N/M˜(2N/M)−1}, . . . , {x[n],n=(M−1)N/M˜N−1} can be obtained to represent M disjoint time domainsignals. Using the symmetric characteristics to process the abovesubblocks, the method of high-order PAPR reduction for OFDM signals canbe obtained, as shown in FIG. 8. First, the N-IFFT 801 is performed onthe frequency domain signal X[k] of length N to obtain the time domainsignal x[n] of length N. The adjacent partitioning is performed on x[n]to obtain M disjoint subblocks having the length N/M, where M is a powerof 2, great or equal to 8, and N/M is an integer greater than 1, as instep 803. In step 805, a transformer is used to transform the M disjointsubblocks into M sub-sequences z_(l)[n], each having the length N/M,l=1, 2, . . . , M and n=0, 1, . . . , (N/M)−1. In step 807, thefixed-phase rotation is used to rotate the M sub-sequences z_(l)[n] toobtain M sub-seuqneces {tilde over (z)}_(l)[n], each having the lengthof N/M, =1, 2, . . . , M and n=0, 1, . . . , (N/M)−1. Finally, in step809, the M sub-sequences {tilde over (z)}_(l)[n], after phase rotationand phase optimization, are added to form a complete N-point transmittedsignal {tilde over (x)}[n], where n=0, 1, . . . , N−1.

In step 807, not all of the fixed-phase rotations require a complexmultiplier, therefore, FIG. 9 shows a schematic view of a preferredembodiment of a high-order PAPR reduction apparatus of the presentinvention. The apparatus includes an N-IFFT 801, a de-multiplexer 903,two sets of memories 905, 909, a transformer 907, at most M complexmultipliers (for fixed-phase rotations), and an adder 911. N-IFFT 801transforms the input frequency domain signal X[k] into the time domainsignal x[n]. De-multiplexer 903 partitions x[n] into M adjacent, yetdisjoint, subblocks {x[n], n=0˜(N/M)−1}{grave over ( )}{x[n],n=N/M˜(2N/M)−1}, . . . , {x[n], n=(M−1)N/M˜N−1} of the identical lengthN/M, stored in memory 905. The M subblocks are transformed bytransformer 907 into M sub-sequences z_(l)[n], each having length N/M,l=1, 2, . . . , M and n=0, 1, . . . , (N/M)−1, also stored in memory905. Some of the M sub-sequences z_(l)[n] are passed through complexmultiplier e^(jθl) to form another sub-sequences {tilde over(z)}_(l)[n], stored in memory 909, where l is between 1 and M, n=0, 1, .. . , (N/M)−1. The next step is to select sub-sequence z[n] or {tildeover (z)}_(l)[n] for phase rotation according to different time period.Finally, adder 911 adds them to form a complete N-point signal {tildeover (x)}[n], where n=0, 1, . . . , N−1.

Using M=8 as an example, FIG. 10 shows a schematic view. The inputsignal X[k] is transformed into x[n] by N-IFFT 801. De-multiplezer 1003takes x[n], n=0, 1, . . . , N−1 and partitions into eight subblocks{x[n], n=0˜(N/8)−1}, {x[n], n=N/8˜(2N/8)−1}, . . . , {x[n], n=7N/8˜N−1},each having length N/8, stored in memory 1005. Transformer 1007transforms the eight subblocks into eight sub-sequences z₁[n], z₂[n], .. . , z₈[n]. According equations (6), (7), and (8), the followingequations can be obtained:

$\begin{matrix}{{z_{1}\lbrack n\rbrack} = {{x\lbrack n\rbrack} + {x\left\lbrack \left( \left( {n - {N/2}} \right) \right)_{N} \right\rbrack} + {x\left\lbrack \left( \left( {n - {N/4}} \right) \right)_{N} \right\rbrack} + {x\left\lbrack \left( \left( {n - {3{N/4}}} \right) \right)_{N} \right\rbrack} + {x\left\lbrack \left( \left( {n - {N/8}} \right) \right)_{N} \right\rbrack} + {x\left\lbrack \left( \left( {n - {5{N/8}}} \right) \right)_{N} \right\rbrack} + {x\left\lbrack \left( \left( {n - {3{N/8}}} \right) \right)_{N} \right\rbrack} + {x\left\lbrack \left( \left( {n - {7{N/8}}} \right) \right)_{N} \right\rbrack}}} & \left( {9a} \right) \\{{z_{5}\lbrack n\rbrack} = {{x\lbrack n\rbrack} + {x\left\lbrack \left( \left( {n - {N/2}} \right) \right)_{N} \right\rbrack} + {x\left\lbrack \left( \left( {n - {N/4}} \right) \right)_{N} \right\rbrack} + {x\left\lbrack \left( \left( {n - {3{N/4}}} \right) \right)_{N} \right\rbrack} - {x\left\lbrack \left( \left( {n - {N/8}} \right) \right)_{N} \right\rbrack} - {x\left\lbrack \left( \left( {n - {5{N/8}}} \right) \right)_{N} \right\rbrack} - {x\left\lbrack \left( \left( {n - {3{N/8}}} \right) \right)_{N} \right\rbrack} - {x\left\lbrack \left( \left( {n - {7{N/8}}} \right) \right)_{N} \right\rbrack}}} & \left( {9b} \right) \\{{z_{3}\lbrack n\rbrack} = {{x\lbrack n\rbrack} + {x\left\lbrack \left( \left( {n - {N/2}} \right) \right)_{N} \right\rbrack} - {x\left\lbrack \left( \left( {n - {N/4}} \right) \right)_{N} \right\rbrack} - {x\left\lbrack \left( \left( {n - {3{N/4}}} \right) \right)_{N} \right\rbrack} + {{jx}\left\lbrack \left( \left( {n - {N/8}} \right) \right)_{N} \right\rbrack} + {{jx}\left\lbrack \left( \left( {n - {5{N/8}}} \right) \right)_{N} \right\rbrack} - {{jx}\left\lbrack \left( \left( {n - {3{N/8}}} \right) \right)_{N} \right\rbrack} - {{jx}\left\lbrack \left( \left( {n - {7{N/8}}} \right) \right)_{N} \right\rbrack}}} & \left( {9c} \right) \\{{z_{7}\lbrack n\rbrack} = {{x\lbrack n\rbrack} + {x\left\lbrack \left( \left( {n - {N/2}} \right) \right)_{N} \right\rbrack} - {x\left\lbrack \left( \left( {n - {N/4}} \right) \right)_{N} \right\rbrack} + {x\left\lbrack \left( \left( {n - {3{N/4}}} \right) \right)_{N} \right\rbrack} - {{jx}\left\lbrack \left( \left( {n - {N/8}} \right) \right)_{N} \right\rbrack} - {{jx}\left\lbrack \left( \left( {n - {5{N/8}}} \right) \right)_{N} \right\rbrack} + {{jx}\left\lbrack \left( \left( {n - {3{N/8}}} \right) \right)_{N} \right\rbrack} + {{jx}\left\lbrack \left( \left( {n - {7{N/8}}} \right) \right)_{N} \right\rbrack}}} & \left( {9d} \right) \\{{z_{2}\lbrack n\rbrack} = {{x\lbrack n\rbrack} - {x\left\lbrack \left( \left( {n - {N/2}} \right) \right)_{N} \right\rbrack} + {{jx}\left\lbrack \left( \left( {n - {N/4}} \right) \right)_{N} \right\rbrack} - {{jx}\left\lbrack \left( \left( {n - {3{N/4}}} \right) \right)_{N} \right\rbrack} + {{\mathbb{e}}^{j\frac{\pi}{4}}\left\{ {{x\left\lbrack \left( \left( {n - {N/8}} \right) \right)_{N} \right\rbrack} - {x\left\lbrack \left( \left( {n - {5{N/8}}} \right) \right)_{N} \right\rbrack} + {{jx}\left\lbrack \left( \left( {n - {3{N/8}}} \right) \right)_{N} \right\rbrack} - {{jx}\left\lbrack \left( \left( {n - {7{N/8}}} \right) \right)_{N} \right\rbrack}} \right\}}}} & \left( {9e} \right) \\{{z_{6}\lbrack n\rbrack} = {{x\lbrack n\rbrack} - {x\left\lbrack \left( \left( {n - {N/2}} \right) \right)_{N} \right\rbrack} + {{jx}\left\lbrack \left( \left( {n - {N/4}} \right) \right)_{N} \right\rbrack} - {{jx}\left\lbrack \left( \left( {n - {3{N/4}}} \right) \right)_{N} \right\rbrack} + {{\mathbb{e}}^{j\frac{\pi}{4}}\left\{ {{- {x\left\lbrack \left( \left( {n - {N/8}} \right) \right)_{N} \right\rbrack}} + {x\left\lbrack \left( \left( {n - {5{N/8}}} \right) \right)_{N} \right\rbrack} - {{jx}\left\lbrack \left( \left( {n - {3{N/8}}} \right) \right)_{N} \right\rbrack} + {{jx}\left\lbrack \left( \left( {n - {7{N/8}}} \right) \right)_{N} \right\rbrack}} \right\}}}} & \left( {9f} \right) \\{{z_{4}\lbrack n\rbrack} = {{x\lbrack n\rbrack} - {x\left\lbrack \left( \left( {n - {N/2}} \right) \right)_{N} \right\rbrack} - {{jx}\left\lbrack \left( \left( {n - {N/4}} \right) \right)_{N} \right\rbrack} + {{jx}\left\lbrack \left( \left( {n - {3{N/4}}} \right) \right)_{N} \right\rbrack} + {{\mathbb{e}}^{j\frac{\pi}{4}}\left\{ {{{jx}\left\lbrack \left( \left( {n - {N/8}} \right) \right)_{N} \right\rbrack} - {{jx}\left\lbrack \left( \left( {n - {5{N/8}}} \right) \right)_{N} \right\rbrack} + {x\left\lbrack \left( \left( {n - {3{N/8}}} \right) \right)_{N} \right\rbrack} - {x\left\lbrack \left( \left( {n - {7{N/8}}} \right) \right)_{N} \right\rbrack}} \right\}}}} & \left( {9g} \right) \\{{z_{8}\lbrack n\rbrack} = {{x\lbrack n\rbrack} - {x\left\lbrack \left( \left( {n - {N/2}} \right) \right)_{N} \right\rbrack} - {{jx}\left\lbrack \left( \left( {n - {N/4}} \right) \right)_{N} \right\rbrack} + {{jx}\left\lbrack \left( \left( {n - {3{N/4}}} \right) \right)_{N} \right\rbrack} + {{\mathbb{e}}^{j\frac{\pi}{4}}\left\{ {{- {{jx}\left\lbrack \left( \left( {n - {N/8}} \right) \right)_{N} \right\rbrack}} + {{jx}\left\lbrack \left( \left( {n - {5{N/8}}} \right) \right)_{N} \right\rbrack} - {x\left\lbrack \left( \left( {n - {3{N/8}}} \right) \right)_{N} \right\rbrack} + {x\left\lbrack \left( \left( {n - {7{N/8}}} \right) \right)_{N} \right\rbrack}} \right\}}}} & \left( {9h} \right)\end{matrix}$Accordingly, a schematic view of transformer 1007 of FIG. 10 can beshown in FIG. 11. As shown in FIG. 11, transformer 1007 uses 24 adders,3 imagery j multipliers, and 2 complex multipliers to implementequations (9a)-(9h). It has a similar structure as 8-IFFT. It can beproven that the sub-sequences z₁[n], z₂[n], . . . , z₈[n] in equations(9a)-(9h) are identical to the time sequences x₁[n], x₂[n], . . . ,x₈[n] obtained from the original PTS method shown in FIG. 2. Therefore,equation (5) can be rewritten as:

$\begin{matrix}{{{\overset{\sim}{x}\lbrack n\rbrack} = {\sum\limits_{l = 1}^{8}{b_{l}{z_{l}\lbrack n\rbrack}}}},\mspace{14mu}{n = 0},1,\ldots\mspace{11mu},{N - 1}} & (10)\end{matrix}$Assuming equations (11a)-(11d) as follows:

$\begin{matrix}{{{\overset{\sim}{z}}_{2}\lbrack n\rbrack} = {{\mathbb{e}}^{j\frac{\pi}{4}} \cdot {z_{2}\lbrack n\rbrack}}} & \left( {11a} \right) \\{{{\overset{\sim}{z}}_{4}\lbrack n\rbrack} = {{\mathbb{e}}^{j\frac{\pi}{4}} \cdot {z_{4}\lbrack n\rbrack}}} & \left( {11b} \right) \\{{{\overset{\sim}{z}}_{6}\lbrack n\rbrack} = {{\mathbb{e}}^{j\frac{\pi}{4}} \cdot {z_{6}\lbrack n\rbrack}}} & \left( {11c} \right) \\{{{\overset{\sim}{z}}_{8}\lbrack n\rbrack} = {{\mathbb{e}}^{j\frac{\pi}{4}} \cdot {z_{8}\lbrack n\rbrack}}} & \left( {11d} \right)\end{matrix}$and using equations (9a)-(9h) and equations (11a)-(11d), the followingsymmetry can be obtained, that is the relationship between thesub-sequences {z₁[((n+pN/8))_(N)], z₂[((n+pN/8))_(N)], . . . ,z₈[((n+pN/8))_(N)]} having different time shift and the sub-sequences{z₁[n], z₂[n], . . . , z₈[n]}.

$\begin{matrix}{{{p = 1},{n = {{{pN}/{\left. 8 \right.\sim\left\lbrack {\left( {p + 1} \right){N/8}} \right\rbrack}} - {1\text{:}}}}}\left\{ \begin{matrix}{{z_{1}\left\lbrack \left( \left( {n + \frac{N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{1}\left\lbrack \left( \left( {n - \frac{7N}{8}} \right) \right)_{N} \right\rbrack} = {z_{1}\lbrack n\rbrack}}} \\{{z_{2}\left\lbrack \left( \left( {n + \frac{N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{2}\left\lbrack \left( \left( {n - \frac{7N}{8}} \right) \right)_{N} \right\rbrack} = {{{\mathbb{e}}^{j\frac{\pi}{4}}{z_{2}\lbrack n\rbrack}} = {{\overset{\sim}{z}}_{2}\lbrack n\rbrack}}}} \\{{z_{3}\left\lbrack \left( \left( {n + \frac{N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{3}\left\lbrack \left( \left( {n - \frac{7N}{8}} \right) \right)_{N} \right\rbrack} = {{jz}_{3}\lbrack n\rbrack}}} \\{{z_{4}\left\lbrack \left( \left( {n + \frac{N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{4}\left\lbrack \left( \left( {n - \frac{7N}{8}} \right) \right)_{N} \right\rbrack} = {{{j\mathbb{e}}^{j\frac{\pi}{4}}{z_{4}\lbrack n\rbrack}} = {j{{\overset{\sim}{z}}_{4}\lbrack n\rbrack}}}}} \\{{z_{5}\left\lbrack \left( \left( {n + \frac{N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{5}\left\lbrack \left( \left( {n - \frac{7N}{8}} \right) \right)_{N} \right\rbrack} = {- {z_{5}\lbrack n\rbrack}}}} \\{{z_{6}\left\lbrack \left( \left( {n + \frac{N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{6}\left\lbrack \left( \left( {n - \frac{7N}{8}} \right) \right)_{N} \right\rbrack} = {{{- {\mathbb{e}}^{j\frac{\pi}{4}}}{z_{6}\lbrack n\rbrack}} = {- {{\overset{\sim}{z}}_{6}\lbrack n\rbrack}}}}} \\{{z_{7}\left\lbrack \left( \left( {n + \frac{N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{7}\left\lbrack \left( \left( {n - \frac{7N}{8}} \right) \right)_{N} \right\rbrack} = {- {{jz}_{7}\lbrack n\rbrack}}}} \\{{z_{8}\left\lbrack \left( \left( {n + \frac{N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{8}\left\lbrack \left( \left( {n - \frac{7N}{8}} \right) \right)_{N} \right\rbrack} = {{{- {j\mathbb{e}}^{j\frac{\pi}{4}}}{z_{8}\lbrack n\rbrack}} = {{- j}{{\overset{\sim}{z}}_{8}\lbrack n\rbrack}}}}}\end{matrix} \right.} & (12) \\{{{p = 2},{n = {{{pN}/{\left. 8 \right.\sim\left\lbrack {\left( {p + 1} \right){N/8}} \right\rbrack}} - {1\text{:}}}}}\left\{ \begin{matrix}{{z_{1}\left\lbrack \left( \left( {n + \frac{2N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{1}\left\lbrack \left( \left( {n - \frac{6N}{8}} \right) \right)_{N} \right\rbrack} = {z_{1}\lbrack n\rbrack}}} \\{{z_{2}\left\lbrack \left( \left( {n + \frac{2N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{2}\left\lbrack \left( \left( {n - \frac{6N}{8}} \right) \right)_{N} \right\rbrack} = {{jz}_{2}\lbrack n\rbrack}}} \\{{z_{3}\left\lbrack \left( \left( {n + \frac{2N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{3}\left\lbrack \left( \left( {n - \frac{6N}{8}} \right) \right)_{N} \right\rbrack} = {- {z_{3}\lbrack n\rbrack}}}} \\{{z_{4}\left\lbrack \left( \left( {n + \frac{2N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{4}\left\lbrack \left( \left( {n - \frac{6N}{8}} \right) \right)_{N} \right\rbrack} = {- {{jz}_{4}\lbrack n\rbrack}}}} \\{{z_{5}\left\lbrack \left( \left( {n + \frac{2N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{5}\left\lbrack \left( \left( {n - \frac{6N}{8}} \right) \right)_{N} \right\rbrack} = {z_{5}\lbrack n\rbrack}}} \\{{z_{6}\left\lbrack \left( \left( {n + \frac{2N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{6}\left\lbrack \left( \left( {n - \frac{6N}{8}} \right) \right)_{N} \right\rbrack} = {{jz}_{6}\lbrack n\rbrack}}} \\{{z_{7}\left\lbrack \left( \left( {n + \frac{2N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{7}\left\lbrack \left( \left( {n - \frac{6N}{8}} \right) \right)_{N} \right\rbrack} = {- {z_{7}\lbrack n\rbrack}}}} \\{{z_{8}\left\lbrack \left( \left( {n + \frac{2N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{8}\left\lbrack \left( \left( {n - \frac{6N}{8}} \right) \right)_{N} \right\rbrack} = {- {{jz}_{8}\lbrack n\rbrack}}}}\end{matrix} \right.} & (13) \\{{{p = 3},{n = {{{pN}/{\left. 8 \right.\sim\left\lbrack {\left( {p + 1} \right){N/8}} \right\rbrack}} - {1\text{:}}}}}\left\{ \begin{matrix}{{z_{1}\left\lbrack \left( \left( {n + \frac{3N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{1}\left\lbrack \left( \left( {n - \frac{5N}{8}} \right) \right)_{N} \right\rbrack} = {z_{1}\lbrack n\rbrack}}} \\{{z_{2}\left\lbrack \left( \left( {n + \frac{3N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{2}\left\lbrack \left( \left( {n - \frac{5N}{8}} \right) \right)_{N} \right\rbrack} = {{{j\mathbb{e}}^{j\frac{\pi}{4}}{z_{2}\lbrack n\rbrack}} = {{\overset{\sim}{jz}}_{2}\lbrack n\rbrack}}}} \\{{z_{3}\left\lbrack \left( \left( {n + \frac{3N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{3}\left\lbrack \left( \left( {n - \frac{5N}{8}} \right) \right)_{N} \right\rbrack} = {- {{jz}_{3}\lbrack n\rbrack}}}} \\{{z_{4}\left\lbrack \left( \left( {n + \frac{3N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{4}\left\lbrack \left( \left( {n - \frac{5N}{8}} \right) \right)_{N} \right\rbrack} = {{{\mathbb{e}}^{j\frac{\pi}{4}}{z_{4}\lbrack n\rbrack}} = {{\overset{\sim}{z}}_{4}\lbrack n\rbrack}}}} \\{{z_{5}\left\lbrack \left( \left( {n + \frac{3N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{5}\left\lbrack \left( \left( {n - \frac{5N}{8}} \right) \right)_{N} \right\rbrack} = {- {z_{5}\lbrack n\rbrack}}}} \\{{z_{6}\left\lbrack \left( \left( {n + \frac{3N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{6}\left\lbrack \left( \left( {n - \frac{5N}{8}} \right) \right)_{N} \right\rbrack} = {{{- {j\mathbb{e}}^{j\frac{\pi}{4}}}{z_{6}\lbrack n\rbrack}} = {{- j}{{\overset{\sim}{z}}_{6}\lbrack n\rbrack}}}}} \\{{z_{7}\left\lbrack \left( \left( {n + \frac{3N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{7}\left\lbrack \left( \left( {n - \frac{5N}{8}} \right) \right)_{N} \right\rbrack} = {{jz}_{7}\lbrack n\rbrack}}} \\{{z_{8}\left\lbrack \left( \left( {n + \frac{3N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{8}\left\lbrack \left( \left( {n - \frac{5N}{8}} \right) \right)_{N} \right\rbrack} = {{{- {\mathbb{e}}^{j\frac{\pi}{4}}}{z_{8}\lbrack n\rbrack}} = {- {{\overset{\sim}{z}}_{8}\lbrack n\rbrack}}}}}\end{matrix} \right.} & (14) \\{{{p = 4},{n = {{{pN}/{\left. 8 \right.\sim\left\lbrack {\left( {p + 1} \right){N/8}} \right\rbrack}} - {1\text{:}}}}}\left\{ \begin{matrix}{{z_{1}\left\lbrack \left( \left( {n + \frac{4N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{1}\left\lbrack \left( \left( {n - \frac{4N}{8}} \right) \right)_{N} \right\rbrack} = {z_{1}\lbrack n\rbrack}}} \\{{z_{2}\left\lbrack \left( \left( {n + \frac{4N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{2}\left\lbrack \left( \left( {n - \frac{4N}{8}} \right) \right)_{N} \right\rbrack} = {- {z_{2}\lbrack n\rbrack}}}} \\{{z_{3}\left\lbrack \left( \left( {n + \frac{4N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{3}\left\lbrack \left( \left( {n - \frac{4N}{8}} \right) \right)_{N} \right\rbrack} = {z_{3}\lbrack n\rbrack}}} \\{{z_{4}\left\lbrack \left( \left( {n + \frac{4N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{4}\left\lbrack \left( \left( {n - \frac{4N}{8}} \right) \right)_{N} \right\rbrack} = {- {z_{4}\lbrack n\rbrack}}}} \\{{z_{5}\left\lbrack \left( \left( {n + \frac{4N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{5}\left\lbrack \left( \left( {n - \frac{4N}{8}} \right) \right)_{N} \right\rbrack} = {z_{5}\lbrack n\rbrack}}} \\{{z_{6}\left\lbrack \left( \left( {n + \frac{4N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{6}\left\lbrack \left( \left( {n - \frac{4N}{8}} \right) \right)_{N} \right\rbrack} = {- {z_{6}\lbrack n\rbrack}}}} \\{{z_{7}\left\lbrack \left( \left( {n + \frac{4N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{7}\left\lbrack \left( \left( {n - \frac{4N}{8}} \right) \right)_{N} \right\rbrack} = {z_{7}\lbrack n\rbrack}}} \\{{z_{8}\left\lbrack \left( \left( {n + \frac{4N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{8}\left\lbrack \left( \left( {n - \frac{4N}{8}} \right) \right)_{N} \right\rbrack} = {- {z_{8}\lbrack n\rbrack}}}}\end{matrix} \right.} & (15) \\{{{p = 5},{n = {{{pN}/{\left. 8 \right.\sim\left\lbrack {\left( {p + 1} \right){N/8}} \right\rbrack}} - {1\text{:}}}}}\left\{ \begin{matrix}{{z_{1}\left\lbrack \left( \left( {n + \frac{5N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{1}\left\lbrack \left( \left( {n - \frac{3N}{8}} \right) \right)_{N} \right\rbrack} = {z_{1}\lbrack n\rbrack}}} \\{{z_{2}\left\lbrack \left( \left( {n + \frac{5N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{2}\left\lbrack \left( \left( {n - \frac{3N}{8}} \right) \right)_{N} \right\rbrack} = {{{- {\mathbb{e}}^{j\frac{\pi}{4}}}{z_{2}\lbrack n\rbrack}} = {- {{\overset{\sim}{z}}_{2}\lbrack n\rbrack}}}}} \\{{z_{3}\left\lbrack \left( \left( {n + \frac{5N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{3}\left\lbrack \left( \left( {n - \frac{3N}{8}} \right) \right)_{N} \right\rbrack} = {{jz}_{3}\lbrack n\rbrack}}} \\{{z_{4}\left\lbrack \left( \left( {n + \frac{5N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{4}\left\lbrack \left( \left( {n - \frac{3N}{8}} \right) \right)_{N} \right\rbrack} = {{{- {j\mathbb{e}}^{j\frac{\pi}{4}}}{z_{4}\lbrack n\rbrack}} = {{- j}{{\overset{\sim}{z}}_{4}\lbrack n\rbrack}}}}} \\{{z_{5}\left\lbrack \left( \left( {n + \frac{5N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{5}\left\lbrack \left( \left( {n - \frac{3N}{8}} \right) \right)_{N} \right\rbrack} = {- {z_{5}\lbrack n\rbrack}}}} \\{{z_{6}\left\lbrack \left( \left( {n + \frac{5N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{6}\left\lbrack \left( \left( {n - \frac{3N}{8}} \right) \right)_{N} \right\rbrack} = {{{\mathbb{e}}^{j\frac{\pi}{4}}{z_{6}\lbrack n\rbrack}} = {{\overset{\sim}{z}}_{6}\lbrack n\rbrack}}}} \\{{z_{7}\left\lbrack \left( \left( {n + \frac{5N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{7}\left\lbrack \left( \left( {n - \frac{3N}{8}} \right) \right)_{N} \right\rbrack} = {- {{jz}_{7}\lbrack n\rbrack}}}} \\{{z_{8}\left\lbrack \left( \left( {n + \frac{5N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{8}\left\lbrack \left( \left( {n - \frac{3N}{8}} \right) \right)_{N} \right\rbrack} = {{{j\mathbb{e}}^{j\frac{\pi}{4}}{z_{8}\lbrack n\rbrack}} = {j{{\overset{\sim}{z}}_{8}\lbrack n\rbrack}}}}}\end{matrix} \right.} & (16) \\{{{p = 6},{n = {{{pN}/{\left. 8 \right.\sim\left\lbrack {\left( {p + 1} \right){N/8}} \right\rbrack}} - {1\text{:}}}}}\left\{ \begin{matrix}{{z_{1}\left\lbrack \left( \left( {n + \frac{6N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{1}\left\lbrack \left( \left( {n - \frac{2N}{8}} \right) \right)_{N} \right\rbrack} = {z_{1}\lbrack n\rbrack}}} \\{{z_{2}\left\lbrack \left( \left( {n + \frac{6N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{2}\left\lbrack \left( \left( {n - \frac{2N}{8}} \right) \right)_{N} \right\rbrack} = {- {{jz}_{2}\lbrack n\rbrack}}}} \\{{z_{3}\left\lbrack \left( \left( {n + \frac{6N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{3}\left\lbrack \left( \left( {n - \frac{2N}{8}} \right) \right)_{N} \right\rbrack} = {- {z_{3}\lbrack n\rbrack}}}} \\{{z_{4}\left\lbrack \left( \left( {n + \frac{6N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{4}\left\lbrack \left( \left( {n - \frac{2N}{8}} \right) \right)_{N} \right\rbrack} = {{jz}_{4}\lbrack n\rbrack}}} \\{{z_{5}\left\lbrack \left( \left( {n + \frac{6N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{5}\left\lbrack \left( \left( {n - \frac{2N}{8}} \right) \right)_{N} \right\rbrack} = {z_{5}\lbrack n\rbrack}}} \\{{z_{6}\left\lbrack \left( \left( {n + \frac{6N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{6}\left\lbrack \left( \left( {n - \frac{2N}{8}} \right) \right)_{N} \right\rbrack} = {- {{jz}_{6}\lbrack n\rbrack}}}} \\{{z_{7}\left\lbrack \left( \left( {n + \frac{6N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{7}\left\lbrack \left( \left( {n - \frac{2N}{8}} \right) \right)_{N} \right\rbrack} = {- {z_{7}\lbrack n\rbrack}}}} \\{{z_{8}\left\lbrack \left( \left( {n + \frac{6N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{8}\left\lbrack \left( \left( {n - \frac{2N}{8}} \right) \right)_{N} \right\rbrack} = {{jz}_{8}\lbrack n\rbrack}}}\end{matrix} \right.} & (17) \\{{{p = 7},{n = {{{pN}/{\left. 8 \right.\sim\left\lbrack {\left( {p + 1} \right){N/8}} \right\rbrack}} - {1\text{:}}}}}\left\{ \begin{matrix}{{z_{1}\left\lbrack \left( \left( {n + \frac{7N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{1}\left\lbrack \left( \left( {n - \frac{N}{8}} \right) \right)_{N} \right\rbrack} = {z_{1}\lbrack n\rbrack}}} \\{{z_{2}\left\lbrack \left( \left( {n + \frac{7N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{2}\left\lbrack \left( \left( {n - \frac{N}{8}} \right) \right)_{N} \right\rbrack} = {{{- {j\mathbb{e}}^{j\frac{\pi}{4}}}{z_{2}\lbrack n\rbrack}} = {{- j}{{\overset{\sim}{z}}_{2}\lbrack n\rbrack}}}}} \\{{z_{3}\left\lbrack \left( \left( {n + \frac{7N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{3}\left\lbrack \left( \left( {n - \frac{N}{8}} \right) \right)_{N} \right\rbrack} = {- {{jz}_{3}\lbrack n\rbrack}}}} \\{{z_{4}\left\lbrack \left( \left( {n + \frac{7N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{4}\left\lbrack \left( \left( {n - \frac{N}{8}} \right) \right)_{N} \right\rbrack} = {{{- {\mathbb{e}}^{j\frac{\pi}{4}}}{z_{4}\lbrack n\rbrack}} = {- {{\overset{\sim}{z}}_{4}\lbrack n\rbrack}}}}} \\{{z_{5}\left\lbrack \left( \left( {n + \frac{7N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{5}\left\lbrack \left( \left( {n - \frac{N}{8}} \right) \right)_{N} \right\rbrack} = {- {z_{5}\lbrack n\rbrack}}}} \\{{z_{6}\left\lbrack \left( \left( {n + \frac{7N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{6}\left\lbrack \left( \left( {n - \frac{N}{8}} \right) \right)_{N} \right\rbrack} = {{{j\mathbb{e}}^{j\frac{\pi}{4}}{z_{6}\lbrack n\rbrack}} = {j{{\overset{\sim}{z}}_{6}\lbrack n\rbrack}}}}} \\{{z_{7}\left\lbrack \left( \left( {n + \frac{7N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{7}\left\lbrack \left( \left( {n - \frac{N}{8}} \right) \right)_{N} \right\rbrack} = {{jz}_{7}\lbrack n\rbrack}}} \\{{z_{8}\left\lbrack \left( \left( {n + \frac{7N}{8}} \right) \right)_{N} \right\rbrack} = {{z_{8}\left\lbrack \left( \left( {n - \frac{N}{8}} \right) \right)_{N} \right\rbrack} = {{{\mathbb{e}}^{j\frac{\pi}{4}}{z_{8}\lbrack n\rbrack}} = {{\overset{\sim}{z}}_{8}\lbrack n\rbrack}}}}\end{matrix} \right.} & (18)\end{matrix}$It can be seen from equations (12)-(18) that transformer 1007 only needsto form N/8-length sub-sequences z₁[n], z₂[n], z₃[n], z₄[n], z₅[n],z₆[n], z₇[n], z₈[n] and obtain {tilde over (z)}₂[n], {tilde over(z)}₄[n], {tilde over (z)}₆[n], {tilde over (z)}₈[n], n=0, . . .(N/8)−1. As the eight subblocks {x[n], n=0, . . . , (N/8)−1}, {x[n],n=N/8, . . . , 2N/8}−1}, . . . , {x[n], n=7N/8, . . . , N−1} are nolonger needed, the eight memory blocks having N/8 units each in memory1005 can be released to store z₁[n], z₂[n], . . . , z₈[n]′n=0, . . . ,(N/8)−1. In other words, the required memory is 8×(N/8)+4×(N/8)=3N/2units. Finally substituting equations (12)-(18) into equation (10), thefollowing two results can be obtained:

-   Result 1: p=0, 2, 4, 6, n=0, . . . , (N/8)−1

$\begin{matrix}{{\overset{\sim}{x}\left\lbrack {n + \frac{pN}{8}} \right\rbrack} = {\sum\limits_{l = 1}^{8}{{\overset{\sim}{b}}_{l} \cdot z_{l}}}} & \text{(19a)}\end{matrix}$

where the phase rotation parameter {tilde over (b)}_(l) is shown in FIG.12( a).

-   Result 2: p=1, 3, 5, 7, n=0, . . . , (N/8)−1

$\begin{matrix}{{\overset{\sim}{x}\left\lbrack {n + \frac{pN}{8}} \right\rbrack} = {{\sum\limits_{l = 1}^{4}{{\overset{\sim}{b}}_{{2l} - 1} \cdot z_{{2l} - 1}}} + {\sum\limits_{l = 1}^{4}{{\overset{\sim}{b}}_{2l} \cdot {\overset{\sim}{z}}_{2l}}}}} & \text{(19b)}\end{matrix}$

where the phase rotation parameter {tilde over (b)}_(l) is shown in FIG.12( b).

Refer to FIG. 10, after the phase rotation of the eight sub-sequences{z₁[n], z₂[n], . . . , z₈[n]}, n=0, . . . , (N/8)−1, adder 1011 addsthem to form the N/8-length transmitted signal {{tilde over (x)}[0],{tilde over (x)}[1], . . . , [(N/8)−1]}. By using different phaserotation parameter {tilde over (b)}_(l), the eight sub-sequences {z₁[n],z₃[n], z₅[n], z₇[n]} and {{tilde over (z)}₂[n], {tilde over (z)}₄[n],{tilde over (z)}₆[n], {tilde over (z)}₈[n]} can be used to obtain thenext transmitted signal {{tilde over (x)}[N/8], {tilde over(x)}[(N/8)+1], . . . , x[(2N/8)−1]}. By the same way, the entire N-pointtransmitted signal {tilde over (x)}[n] can be obtained.

As shown in FIG. 10 and FIG. 11, when {tilde over (b)}_(l) is +1, −1,+j, or −j, the multiplications that present invention requires come fromN-IFFT 801, transformer 1007 (including two complex multipliers), andfour complex multipliers (for fixed-phase rotations). Therefore, thetotal amount of complex multiplications is(N/2)log₂N+2×(N/8)+4×(N/8)=(N/2)log₂N+(3N/4), and the memory requirementis 3N/2 units.

The following description uses M=8 as an example to explain thesimplified embodiment of the high-order PAPR reduction apparatus of thepresent invention. By losing a slight capability for PAPR reduction, asmaller amount of multiplications and memory requirement can beachieved. FIG. 13 shows a schematic view of the simplified embodiment.The frequency domain sequence X[k] is transformed by N-IFFT 801 intotime domain signal sequence x[n]. De-multiplexer 1003 partitions x[n],n=0, 1, . . . , N−1 into 8 subblocks {x[n], n=0, . . . , (N/8)−1},{x[n], n=N/8, . . . , (2N/8)−1}, . . . , {x[n], n=7N/8, . . . , N−1},each having the length N/8, and stores them into 8 N/8-length blocks inmemory 1005. Substituting equations (9a)-(9h) into equation (10), thefollowing equation can be obtained:

$\begin{matrix}{{\overset{\sim}{x}\lbrack n\rbrack} = {{{x\lbrack n\rbrack} \cdot \left( {b_{1} + b_{5} + b_{3} + b_{7} + b_{2} + b_{6} + b_{4} + b_{8}} \right)} + {{x\left\lbrack \left( \left( {N - \frac{N}{2}} \right) \right)_{N} \right\rbrack} \cdot \left( {b_{1} + b_{5} + b_{3} + b_{7} - b_{2} - b_{6} - b_{4} - b_{8}} \right)} + {{x\left\lbrack \left( \left( {N - \frac{N}{4}} \right) \right)_{N} \right\rbrack} \cdot \left( {b_{1} + b_{5} - b_{3} - b_{7} + {jb}_{2} + {jb}_{6} - {jb}_{4} - {jb}_{8}} \right)} + {{x\left\lbrack \left( \left( {N - \frac{3N}{4}} \right) \right)_{N} \right\rbrack} \cdot \left( {b_{1} + b_{5} - b_{3} - b_{7} - {jb}_{2} - {jb}_{6} + {jb}_{4} + {jb}_{8}} \right)} + {{x\left\lbrack \left( \left( {N - \frac{N}{8}} \right) \right)_{N} \right\rbrack} \cdot \left\{ {b_{1} - b_{5} + {jb}_{3} - {jb}_{7} + {{\mathbb{e}}^{j\frac{\pi}{4}}\left( {b_{2} - b_{6}} \right)} + {{j\mathbb{e}}^{j\frac{\pi}{4}}\left( {b_{4} - b_{8}} \right)}} \right\}} + {{x\left\lbrack \left( \left( {N - \frac{N}{8}} \right) \right)_{N} \right\rbrack} \cdot \left\{ {b_{1} - b_{5} + {jb}_{3} - {jb}_{7} - {{\mathbb{e}}^{j\frac{\pi}{4}}\left( {b_{2} - b_{6}} \right)} - {{j\mathbb{e}}^{j\frac{\pi}{4}}\left( {b_{4} - b_{8}} \right)}} \right\}} + {{x\left\lbrack \left( \left( {N - \frac{3N}{8}} \right) \right)_{N} \right\rbrack} \cdot \left\{ {b_{1} - b_{5} - {jb}_{3} + {jb}_{7} + {{j\mathbb{e}}^{j\frac{\pi}{4}}\left( {b_{2} - b_{6}} \right)} + {{\mathbb{e}}^{j\frac{\pi}{4}}\left( {b_{4} - 8} \right)}} \right\}} + {{x\left\lbrack \left( \left( {N - \frac{7N}{8}} \right) \right)_{N} \right\rbrack} \cdot \left\{ {b_{1} - b_{5} + {jb}_{3} + {jb}_{7} - {{j\mathbb{e}}^{j\frac{\pi}{4}}\left( {b_{2} - b_{6}} \right)} - {{\mathbb{e}}^{j\frac{\pi}{4}}\left( {b_{4} - b_{8}} \right)}} \right\}}}} & (20)\end{matrix}$

Equation (20) shows that when b₂=b₆ and b₄=b₈, it does not requirecomplex multiplication to compute equation (20). Under suchcircumstances, using equations (9a)-(9h) to express equation (20) asfollowing:

$\begin{matrix}{{\overset{\sim}{x}\;\lbrack n\rbrack} = {{b_{1}{z_{1}\lbrack n\rbrack}} + {b_{5}{z_{5}\lbrack n\rbrack}} + {b_{3}{z_{3}\lbrack n\rbrack}} + {b_{7}{z_{7}\lbrack n\rbrack}} + {b_{2} \cdot \left\{ {{w_{2}\lbrack n\rbrack} + {j\;{w_{4}\lbrack n\rbrack}}} \right\}} + {b_{4} \cdot \left\{ {{w_{2}\lbrack n\rbrack} - {j\;{w_{4}\lbrack n\rbrack}}} \right\}}}} & (21) \\{where} & \; \\{{w_{2}\lbrack n\rbrack} = {2 \cdot \left\{ {{x\;\lbrack n\rbrack} - {x\;\left\lbrack \left( \left( {n - \frac{N}{2}} \right) \right)_{N} \right\rbrack}} \right\}}} & \text{(22a)} \\{{w_{4}\lbrack n\rbrack} = {2 \cdot \left\{ {{x\;\left\lbrack \left( \left( {n - \frac{N}{4}} \right) \right)_{N} \right\rbrack} - {x\;\left\lbrack \left( \left( {n - \frac{3N}{4}} \right) \right)_{N} \right\rbrack}} \right\}}} & \text{(22b)}\end{matrix}$Based on equations (21) and (12), the following is obtained:

$\begin{matrix}{{\overset{\sim}{x}\left\lbrack \left( \left( {n + \frac{N}{8}} \right) \right)_{N} \right\rbrack} = {{b_{1}{z_{1}\lbrack n\rbrack}} - {b_{5}{z_{5}\lbrack n\rbrack}} + {{jb}_{3}{z_{3}\lbrack n\rbrack}} - {{jb}_{7}{z_{7}\lbrack n\rbrack}} + {b_{2} \cdot \left\{ {{w_{2}\left\{ \left( \left( {n + \frac{N}{8}} \right) \right)_{N} \right\rbrack} + {{jw}_{4}\left\lbrack \left( \left( {n + \frac{N}{8}} \right) \right)_{N} \right\rbrack}} \right\}} + {b_{4} \cdot \left\{ {{w_{2}\left\lbrack \left( \left( {n + \frac{N}{8}} \right) \right)_{N} \right\rbrack} - {{jw}_{4}\left\lbrack \left( \left( {n + \frac{n}{8}} \right) \right)_{N} \right\rbrack}} \right\}}}} & (23)\end{matrix}$Furthermore, the symmetric relationship can be obtained from equations(22a) and (22b):

$\begin{matrix}\left\{ \begin{matrix}{{w_{2}\left\lbrack \left( \left( {n + \frac{2N}{8}} \right) \right)_{N} \right\rbrack} = {{w_{2}\left\lbrack \left( \left( {n - \frac{6N}{8}} \right) \right)_{N} \right\rbrack} = {- {w_{4}\lbrack n\rbrack}}}} \\{{w_{4}\left\lbrack \left( \left( {n + \frac{2N}{8}} \right) \right)_{N} \right\rbrack} = {{w_{4}\left\lbrack \left( \left( {n - \frac{6N}{8}} \right) \right)_{N} \right\rbrack} = {w_{2}\lbrack n\rbrack}}}\end{matrix} \right. & \text{(24a)} \\\left\{ \begin{matrix}{{w_{2}\left\lbrack \left( \left( {n + \frac{3N}{8}} \right) \right)_{N} \right\rbrack} = {{w_{2}\left\lbrack \left( \left( {n - \frac{5N}{8}} \right) \right)_{N} \right\rbrack} = {- {w_{4}\left\lbrack \left( \left( {n + \frac{N}{8}} \right) \right)_{N} \right\rbrack}}}} \\{{w_{4}\left\lbrack \left( \left( {n + \frac{3N}{8}} \right) \right)_{N} \right\rbrack} = {{w_{4}\left\lbrack \left( \left( {n - \frac{5N}{8}} \right) \right)_{N} \right\rbrack} = {w_{2}\left\lbrack \left( \left( {n + \frac{N}{8}} \right) \right)_{N} \right\rbrack}}}\end{matrix} \right. & \text{(24b)} \\\left\{ \begin{matrix}{{w_{2}\left\lbrack \left( \left( {n + \frac{4N}{8}} \right) \right)_{N} \right\rbrack} = {{w_{2}\left\lbrack \left( \left( {n - \frac{4N}{8}} \right) \right)_{N} \right\rbrack} = {- {w_{2}\lbrack n\rbrack}}}} \\{{w_{4}\left\lbrack \left( \left( {n + \frac{4N}{8}} \right) \right)_{N} \right\rbrack} = {{w_{4}\left\lbrack \left( \left( {n - \frac{4N}{8}} \right) \right)_{N} \right\rbrack} = {- {w_{4}\lbrack n\rbrack}}}}\end{matrix} \right. & \text{(24c)} \\\left\{ \begin{matrix}{{w_{2}\left\lbrack \left( \left( {n + \frac{5N}{8}} \right) \right)_{N} \right\rbrack} = {{w_{2}\left\lbrack \left( \left( {n - \frac{3N}{8}} \right) \right)_{N} \right\rbrack} = {- {w_{2}\left\lbrack \left( \left( {n + \frac{N}{8}} \right) \right)_{N} \right\rbrack}}}} \\{{w_{4}\left\lbrack \left( \left( {n + \frac{5N}{8}} \right) \right)_{N} \right\rbrack} = {{w_{4}\left\lbrack \left( \left( {n - \frac{3N}{8}} \right) \right)_{N} \right\rbrack} = {- {w_{4}\left\lbrack \left( \left( {n + \frac{N}{8}} \right) \right)_{N} \right\rbrack}}}}\end{matrix} \right. & \text{(24d)} \\\left\{ \begin{matrix}{{w_{2}\left\lbrack \left( \left( {n + \frac{6N}{8}} \right) \right)_{N} \right\rbrack} = {{w_{2}\left\lbrack \left( \left( {n - \frac{2N}{8}} \right) \right)_{N} \right\rbrack} = {w_{4}\lbrack n\rbrack}}} \\{{w_{4}\left\lbrack \left( \left( {n + \frac{6N}{8}} \right) \right)_{N} \right\rbrack} = {{w_{4}\left\lbrack \left( \left( {n - \frac{2N}{8}} \right) \right)_{N} \right\rbrack} = {- {w_{2}\lbrack n\rbrack}}}}\end{matrix} \right. & \text{(24e)} \\\left\{ \begin{matrix}{{w_{2}\left\lbrack \left( \left( {n + \frac{7N}{8}} \right) \right)_{N} \right\rbrack} = {{w_{2}\left\lbrack \left( \left( {n - \frac{N}{8}} \right) \right)_{N} \right\rbrack} = {w_{4}\left\lbrack \left( \left( {n + \frac{N}{8}} \right) \right)_{N} \right\rbrack}}} \\{{w_{4}\left\lbrack \left( \left( {n + \frac{7N}{8}} \right) \right)_{N} \right\rbrack} = {{w_{4}\left\lbrack \left( \left( {n - \frac{N}{8}} \right) \right)_{N} \right\rbrack} = {- {w_{2}\left\lbrack \left( \left( {n + \frac{N}{8}} \right) \right)_{N} \right\rbrack}}}}\end{matrix} \right. & \left( {24f} \right)\end{matrix}$

Based on equations (21), (23), and (24a)-(24f), the transmitted signal{tilde over (x)}[n] is composed of sub-sequences {z₁[n], z₃[n], z₅[n],z₇[n]} and {w₂[n], w₄[n], w₂[((n+N/8))_(N)], w₄[((n+N/8))_(N)]}.Therefore, transformer 1307 is to transform the eight N/8-lengthsubblocks {x[n], n=0, . . . , (N/8)−1}, {x[n], n=N/8, . . . , (2N/8)−1},. . . , {x[n], n=7N/8, . . . , N−1} into eight sub-sequences {z₁[n],z₃[n], z₅[n], z₇[n]} and {w₂[n], w₄[n], w₂[((n+N/8))_(N)],w₄[((n+N/8))_(N)]}. According to equations (9a)-(9d), (22a) and (22b), aschematic view of transformer 1307 of FIG. 13 can be shown in FIG. 14.As shown in FIG. 14, transformer 1307 uses 16 adders, a imagery jmultiplier and four real multipliers to implement equations (9a)-(9d),(22a) and (22d). The structure is different from an 8-IFFT. The fourreal multipliers are for the multiplication of real number 2, which canbe simply implemented as a shift operation. As a result, transformer1307 does not require any multipliers.

It can be seen from equations (12)-(18) and (24a)-(24f) that transformer1307 only needs to form N/8-length sub-sequences {z₁[n], z₃[n], z₅[n],z₇[n]}and {w₂[n], w₄[n], w₂[((n+N/8))N], w₄[((n+N/8))_(N)]}, n=0, . . ., (N/8)−1. As the eight subblocks {x[n], n=0, . . . , (N/8)−1}, {x[n],n=N/8, . . . , (2N/8)−1}, . . . , {x[n], n=7N/8, . . . , N−1} are nolonger needed, the eight memory blocks having N/8 units each in memory1005 can be released to store {z₁[n], z₃[n], z₅[n], z₇[n]}and {w₂[n],w₄[n], w₂[((n+N/8))_(N)], w₄[((n+N/8))N]}, n=0, . . . , (N/8)−1. Inother words, the required memory is 8×(N/8)=N units. Finallysubstituting equations (12)-(18) and (24a)-(24f) into equations (21) and(23), the following two results can be obtained:

-   Result 1: p=0, 2, 4, 6, n=0, . . . , (N/8)−1

$\begin{matrix}{\begin{matrix}{{\overset{\sim}{x}\left\lbrack \left( \left( {n + \frac{pN}{8}} \right) \right)_{N} \right\rbrack} = {{{\overset{\sim}{b}}_{1}{z_{1}\lbrack n\rbrack}} + {{\overset{\sim}{b}}_{5}{z_{5}\lbrack n\rbrack}} + {{\overset{\sim}{b}}_{3}{z_{3}\lbrack n\rbrack}} + {{\overset{\sim}{b}}_{7}{z_{7}\lbrack n\rbrack}} +}} \\{{{\overset{\sim}{b}}_{2}{w_{2}\lbrack n\rbrack}} + {{\overset{\sim}{b}}_{4}{w_{2}\lbrack n\rbrack}} + {{\overset{\sim}{b}}_{6}{w_{4}\lbrack n\rbrack}} + {{\overset{\sim}{b}}_{8}{w_{4}\lbrack n\rbrack}}}\end{matrix}\quad} & \text{(25a)}\end{matrix}$

where the phase rotation parameter {tilde over (b)}_(l) is shown in FIG.15( a).

-   Result 2: p=1, 3, 5, 7, n=0, . . . , (N/8)−1

$\begin{matrix}{\begin{matrix}{{\overset{\sim}{x}\left\lbrack \left( \left( {n + \frac{pN}{8}} \right) \right)_{N} \right\rbrack} = {{{\overset{\sim}{b}}_{1}{z_{1}\lbrack n\rbrack}} + {{\overset{\sim}{b}}_{5}{z_{5}\lbrack n\rbrack}} + {{\overset{\sim}{b}}_{3}{z_{3}\lbrack n\rbrack}} + {{\overset{\sim}{b}}_{7}{z_{7}\lbrack n\rbrack}} +}} \\{{{\overset{\sim}{b}}_{2}{w_{2}\left\lbrack \left( \left( {n + \frac{N}{8}} \right) \right)_{N} \right\rbrack}} + {{\overset{\sim}{b}}_{4}{w_{2}\left\lbrack \left( \left( {n + \frac{N}{8}} \right) \right)_{N} \right\rbrack}} +} \\{{{\overset{\sim}{b}}_{6}{w_{4}\left\lbrack \left( \left( {n + \frac{N}{8}} \right) \right)_{N} \right\rbrack}} + {{\overset{\sim}{b}}_{8}{w_{4}\left\lbrack \left( \left( {n + \frac{N}{8}} \right) \right)_{N} \right\rbrack}}}\end{matrix}\quad} & \text{(25b)}\end{matrix}$

where the phase rotation parameter {tilde over (b)}_(l) is shown in FIG.15( b).

Refer to FIG. 13, after the phase rotation of the six sub-sequences{z₁[n], z₃[n], z₅[n], z₇[n]} and {w₂[n], w₄[n]}, n=0, . . . , (N/8)−1,adder 1011 adds to form an N/8-length transmitted signal {{tilde over(x)}[0], {tilde over (x)}[1], . . . , x[(N/8)−1]}. By using differentphase rotation parameter {tilde over (b)}_(l), the six sub-sequences{z₁[n], z₃[n], z₅[n], z₇[n]} and {w₂[((n+N/8))_(N)], w₄[((n+N/8))_(N)]},n=0, . . . , (N/8)−1, can be used to obtain the next transmitted signal{{tilde over (x)}[N/8], {tilde over (x)}[(N/8)+1], . . . , {tilde over(x)}[(2N/8)−1]}. By the same way, the entire N-point transmitted signal{tilde over (x)}[n] can be obtained.

As shown in FIG. 13 and FIG. 14, when {tilde over (b)}_(l) is +1, −1,+j, or −j, the multiplications that the simplified embodiment of thepresent invention requires only come from N-IFFT 801. Therefore, thetotal amount of complex multiplications is (N/2)log₂N, and the memoryrequirement is N units.

FIG. 16 shows the comparison of the amount of computation and the memoryrequirement of the preferred (FIG. 10) and simplified (FIG. 13)embodiments of the present invention and the other three PTS methods,for M=8 and N=64, 256, 1024, and 2048. As can be seen in FIG. 16, theamount of computation and the memory requirement increases rapidly as Nincreases for the original PTS method and the method disclosed by Kang,Kim and Joo. In comparison, the simplified embodiment of the presentinvention and the Samsung's method take the minimum number ofmultiplications (160, 896, 4608, 10240 multiplications, respectively)and require the minimum memory space (64, 256, 1024, 2048 units ofmemory, respectively). However, the PTS method of the present inventiondoes not shorten the length of the OFDM signal, and therefore stillkeeps the features and the advantages of the original OFDM system.

In summary, the present invention uses the interleaved partitioning ofthe subblocks of the PTS method, and uses only an N-IFFT to provide amethod and an apparatus for high-order PAPR reduction for OFDM signal.The simplified embodiment of the present invention only takes (N/2)log₂Nmultiplications and requires only N units of memory space. Furthermore,the present invention keeps the features and the advantages of theoriginal OFDM system.

Although the present invention has been described with reference to thepreferred embodiments, it will be understood that the invention is notlimited to the details described thereof. Various substitutions andmodifications have been suggested in the foregoing description, andothers will occur to those of ordinary skill in the art. Therefore, allsuch substitutions and modifications are intended to be embraced withinthe scope of the invention as defined in the appended claims.

1. A method for high-order peak-to-average power ration (PAPR) reductionfor an OFDM signal, comprising: (a) transforming a frequency domainsignal X[k] of length N by an N-IFFT into a time domain signal x[n] oflength N, where N is the number of useful data in an OFDM symbol, n=0,1, . . . , N−1; (b) partitioning said sequence x[n] of length N into Mdisjoint subblocks, M being a power of 2 and greater or equal to 8, andN/M being an integer greater than 1; (c) transforming said subblocksinto sub-sequences z₁ [n], each having the length N/M, l=1, 2, . . . , Mand n=0, 1, . . . , (N/M)−1; and (d) each said sub-sequence z₁[n] beingprocessed using complex multiplication followed by phase rotation ordirectly using phase rotation, then addition to form a complete N-pointtransmitted signal {tilde over (x)}[n].
 2. The method as claimed inclaim 1, wherein said transformation in said step (c) is equivalent toan M-point inverse fast Fourier transform (IFFT).
 3. The method asclaimed in claim 1, wherein said M is equal to
 8. 4. The method asclaimed in claim 1, wherein said transformation in said step (c) furthercomprises addition, imagery multiplication or complex multiplication. 5.The method as claimed in claim 1, wherein said complex multiplication insaid step (d) is e^(j(2π/M)).
 6. The method as claimed in claim 1,wherein said method requires less than or equal to (N/2)log₂N+(3N/4)complex multiplications.
 7. The method as claimed in claim 1, whereinsaid step (d) further comprises the steps of: (d1) a part of said Msub-sequences z₁[n] being processed using complex multiplicatione^(j(2π/M)) to form another sub-sequence {tilde over (z)}_(l)[n], lbeing between 1 and M, and n=0, 1, . . . , (N/M)−1; and (d2) usingdifferent time period to select sub-sequence z₁[n] or {tilde over(z)}_(l)[n] for phase rotation, and adding to form a complete N-pointtransmitted signal {tilde over (x)}[n].
 8. The method as claimed inclaim 1, wherein in said step (d), said phase rotation parameter is ofthe four possibilities {+1, −1, +j, −j}.
 9. An apparatus for PAPRreduction of an OFDM signal, comprising: an N-point inverse fast Fouriertransform (N-IFFT) for transforming N-length frequency domain signalX[k] into N-length time domain signal x[n], where N is the number ofuseful data in an OFDM symbol, n=0, 1, . . . , N−1; a de-multiplexer forpartitioning said time domain signal x[n] of length N into M disjointsubblocks of identical length N/M, where n=0, 1, . . . , N−1, N is aninteger greater than 1, M is a power of 2 and greater than or equal to8, and N/M is an integer greater than 1; a transformer transforming saidM disjoint subblocks into M N/M-length sub-sequences z[n], where l=1, 2,. . . , M and n=0, 1, . . . , (N/M)−1, for further complexmultiplication or phase rotation; at most M complex multipliersexecuting said complex multiplication on a part of said sub-sequencesz₁[n] to form another sub-sequence {tilde over (z)}_(l)[n], where l isbetween 1 and M, and n=0, 1, . . . , (N/M)−1; two sets of memories, onesaid memory storing said M disjoint subblocks and said M sub-sequencesz₁[n], and the other said memory storing said sub-sequences z₁[n]; andan adder for adding said sub-sequences z₁[n] and {tilde over(z)}_(l)[n], after phase rotation, to obtain a complete N-pointtransmitted signal {tilde over (x)}[n].
 10. The apparatus as claimed inclaim 9, wherein said M is 8, and said transformer is implemented with24 adders, 3 imagery j multipliers and 2 complex multipliers.
 11. Theapparatus as claimed in claim 9, wherein said M is 8, and saidtransformer is implemented with 16 adders, 1 imagery j multipliers and 4real multipliers.
 12. The apparatus as claimed in claim 9, wherein saidapparatus only uses an N-point inverse fast Fourier transform (N-IFFT).13. The apparatus as claimed in claim 9, wherein said two sets ofmemories require less than or equal to 3N/2 units of memory.